Determining Factorial Speed Fast

TL;DR

It is shown that any graph class representable by a finite binary language has at most factorial speed, meaning that its speed function behaves like $2^{\Theta(n\log n)}$, and this criterion is used to classify many graph classes whose speed was previously unknown as factorial.

Abstract

The speed of a graph class $\cal G$ measures how many labeled graphs on $n$ vertices one can find in $\cal G$. This graph class complexity function is explicitly provided on graphclasses.org. However, for many graph classes, their speed status is classified as \emph{unknown}. In this paper, w}\shortversion{W}e show that any graph class representable by a finite binary language has at most factorial speed, meaning that its speed function behaves like $2^{Θ(n\log n)}$, and we use this criterion to classify many graph classes whose speed was previously unknown as factorial. As a consequence, inclusions between several graph classes can now be seen to be proper. We also prove that $k$-letter graphs have exponential speed, i.e., the speed function lies in $2^{Θ(n)}$.

Figures (5)

• Figure 1: Three interval relation patterns: overlap, disjointness and containment.
• Figure 2: Constructing a $2$-interval graph from a $4$-uniform word
• Figure 3: Different patterns in the intersection model of two trapezoids. In $L^{\text{trap}}$, the first two situations correspond to interval overlap and interval containment, expressed by projecting into $\langle 0101,0110\rangle$, while the last two correspond to $\langle 00111100\rangle$.
• Figure 4: A circle-3-gon and a circle-2-gon (circle trapezoid) described by a word, for different positions of the point $c$.
• Figure 5: A boxicity-2 representation of a 2-thin graph. Partition $V^1$ is drawn in blue, partition $V^2$ is drawn in red.

Theorems & Definitions (52)

• definition 1
• definition 2
• theorem 1
• proof
• remark 1
• remark 2
• theorem 2
• proof
• corollary 1
• proof